Converging orifice used to control the discharge rate of spherical particles from a flat floor silo

The effect of the converging orifice geometry in a model silo on the discharge rate of monosized spherical particles was studied experimentally and numerically. The cylindrical container was equipped with interchangeable inserts with converging discharge orifices of various upper diameters in the upper base and a constant lower diameter in the lower base. Plastic PLA beads and agricultural granular materials: wheat, rapeseeds, and linseeds were tested. A series of discrete element method simulations corresponding to the performed experiments was conducted with a largely extended set of experimental discharge conditions. In the case of the constant thickness of the insert, the discharge rate initially increased with an increase in the half cone angle of the converging orifice and then the tendency reversed. In the majority of cases, the discharge rate through the converging orifice was higher than through the hopper with the same orifice diameter.

Recently, reports have been published presenting numerical methods for designing hoppers with a varying contraction rate to maximize the mass discharge rate of granular material. The finite element method 13,14 or the discrete element method 15 with efficiency corroborated by experimental verification 16 are used most often. Some results have shown that MDR can be increased by nearly 140% in a curved hopper, compared to a conical hopper with the same orifice size, hopper height, and silo diameter. Proper silo geometry may allow to control precisely the flow rate of granular material discharging the silo; however, understanding how to manipulate the mass discharge rate requires further research. That may have practical applications in metering, dosing or mixing.
Considering the results of above-mentioned studies, the objective of the reported project was to carry out a systematic study of the flow through a conical converging orifice with various values of thickness and half cone angle. A possibility of replacing the hopper bottom by the flat bottom equipped with converging discharge orifices in silo has been investigated. Motivation for the present study comes from the industrial flow of powders and grains in various devices. The converging parts, e.g. welding neck flanges, are common and important components of many practical apparatus used in the transport and processing of liquids and granular solids 17, 18 .
So far no attempts have been made to use a numerical method for analyzing flow rate of granular materials through a conical converging orifice with various geometry. Therefore, series of the discrete element method simulations, supplemented by laboratory experiments have been performed. The specific appliance was designed for purpose of that project.

Methods and materials
Laboratory testing. The experimental silo has been used to measure the mass discharge rate MDR. The cylindrical flat bottomed container (Fig. 1a) was 150 mm in diameter and 450 mm high. The container wall was made of galvanized steel, while its flat floor was made of plywood. Plastic PLA beads with a diameter of 5.95 mm, d p , and the mass of 0.25 g were used as reference particles. The number of PLA particles in the sample was equal to 14,000. Wheat, rapeseeds, and linseeds were tested as agricultural granular particles (Fig. 1b, Table 1). The frictional parameters of particles were determined with use of the tilting table method ( Table 2). The silo diameter was 25 times bigger than the largest particle diameter, which, according to the findings reported in literature, allowed neglecting the influence of the bin wall [19][20][21] . A repeatable filling procedure was adopted to maintain a similar geometrical bedding structure in subsequent tests. A sieve was placed axially on the top surface of the silo. The measured amount of particles was poured through the sieve. After completion of filling, the top free surface was leveled. The discharge gate was opened and the mass of particles leaving the container was measured until the discharge was completed. Indications of three load cells supporting the silo were used to determine change in the mass of silo and particles during discharge. The change in the mass of discharged particles was   The thicknesses h of the inserts were tested in a range from 0 to 100 mm. The majority of them were normal multiplicities of the particle mean diameter. The lower diameter d 0 ranged from 19 to 55 mm and the upper diameter d 1 ranged from 32.5 to 72 mm, providing the half cone angle ranging between 4 and 90º. The reference lower diameter d 0 of the orifice was 32.5 mm. The flat orifice with d 1 = 32.5 mm (d 0 > d 1 ) served as a reference orifice providing a non-disturbed discharge. The discharge through conical hoppers with the same half cone angle as that of the converging orifice provided additional reference data of the mass discharge rate. The orifice diameter of the hopper was 32.5 mm and the upper diameter was 150 mm. The Hertz-Mindlin no-slip contact model was applied for simulations following the Hertz theory 23 as the default model used in the EDEM software package 24 . The material parameters of the particles were taken to reproduce the properties of the PLA particles: solid density ρ = 2212 kg/m 3 , Young's modulus E = 8.8 GPa, and Poisson's ratio ν = 0.25 25 . The frictional parameters between particles μ p-p = 0.47, between particle and wall μ p-w = 0.49, and between particle and bottom (plastic insert) μ p-b = 0.21, as well as the coefficient of restitution e = 0.3 were determined experimentally. A default value of the rolling friction of 0.01 of EDEM software was applied for simulations. The walls of the silo were modelled with density ρ = 7800 kg/m 3 , Young's modulus E = 200 GPa, and Poisson's ratio ν = 0.25, which were material parameters of the steel.
Particles were generated inside the model silo. The particles were then discharged through a centrally located flat orifice, a converging orifice, or a conical hopper (Fig. 2). The simulations were performed with a time step of 1.6•10 -6 s with use of EDEM software package 24 .
The simulations were performed according to the following scheme of setting the converging orifice parameters:

Results
Discharge scheme No. (1) d 1 = var., α= var., d 0 = const., h = const. The preliminary DEM simulations performed for the flat orifice (d 0 > d 1 ) with the diameter d 1 in the range from 19 to 35 mm indicated that the threshold orifice size providing an undisturbed flow of material from the silo was 32.5 mm. Therefore, in the further study, the lower diameter d 0 = 32.5 mm was applied for the simulations. The DEM simulated relationship between the mass discharge rate MDR and the upper diameter of the converging orifice d 1 for d 0 = 32.5 mm  Figure 3b shows a change in the normalized mass discharge rate (MDR norm. ) with the increasing half cone angle α of the converging orifice. Mass discharge rates were normalized by the mass discharge rate determined for the flat orifice of d 1 = 32.5 mm. For all tested thicknesses, the MDR norm. initially increased with the increasing α. After the maximum was reached at α crit. , the mass flow rate monotonically declined to the MDR obtained for the flat reference orifice (i.e. MDR norm. → 1). The highest maximum of the MDR norm. (> 3) was obtained for α crit. = 4º and h = 100 mm. The maximal values of MDR norm. decreased with the decrease in the thickness of the insert and were noted for the higher half cone angle α crit . For small values of α crit. the maxima MDR norm. obtained for the converging orifice were 5% lower than those obtained for the hopper with the same half cone angle α and the same orifice diameter of 32.5 mm, while the maxima for α > 20º were approximately 10% higher than those for the hopper.
The course of the relationships MDR norm. (α) may be interpreted in the light of the Jenike criterion for the flow pattern in a conical hopper as dependent on the angle of internal friction and on the α value 14,24 . In the case of a steep hopper (low α), a mass flow takes place. After an increase in α to a limiting value, the flow pattern changes into a funnel flow. A further increase in α leads to formation of a stable dead zone with a converging flow identical to that present in a flat floor silo.
The results of the laboratory tests performed for four granular materials discharged from the converging orifice with geometry providing the maximum MDR in the DEM simulations were compared with the numerical  www.nature.com/scientificreports/ results obtained for the same geometry of the converging orifice and for the hopper (Fig. 4). The experimental and numerical results were in reasonable agreement. Both of them showed the same tendency of a decrease in MDR norm. with the α crit. increase. Most of the experimental results were located very close to the results of the simulations performed for the converging orifice. The values of MDR norm. for rapeseeds were lower than those for the other materials, which should be attributed to the over twice larger difference in the size of the seeds. This is consistent with the findings reported by Gella,  (2) the MDR decreasing with α increase for α > α crit. (Fig. 5).   (Fig. 4), the relationships obtained using scheme No. 2 (d 1 = var., d 1 = d 0 + const., h = const., α = const.) followed Beverloo's relationship very well. This means that the relationship MDR(d 1 ) obtained for the converging orifice with α = const. ≤ α crit. followed Beverloo's relationship obtained for the flat orifice.  www.nature.com/scientificreports/ its maximum/plateau and remained almost constant with the further increase in d 0 (Fig. 7a). Substituting the d 0 variable with the corresponding half cone angle α under the condition d 1 = const., it can be observed that the MDR remained almost constant for α ≤ α crit. and decreased with the α increase for α > α crit. (Fig. 7b). Scatter of the MDR illustrated in Fig. 7 as the standard deviation bars disturbed precise determination of α initiating the plateau. The difference in the course of dependencies presented in Figs. 3 and 7 results from applying the different independent x variable: d 1 in Fig. 3a and d 0 Fig. 7a. Additionally, the half cone angle α applied in Fig. 3b and Fig. 7b depends in different way on the variables d 0 and Dense and loose flow through the orifice. Figure 8 shows changes in the mean porosity of the assembly of spherical particles determined in the volume of the orifice of d 0 = 32.5 mm, for insert thickness of 100 mm (Fig. 8a), and 12 mm (Fig. 8b), at detention, after filling, and during commencement of the discharge. Porosity is defined as the ratio of the volume of pores to the volume of the assembly. The time variation of porosity in the volume of the orifice for several values of α has been shown. After filling, the porosity was approximately 48% in static conditions. For the insert with h = 100 mm, the discharge commencement produced a sharp increase in the porosity to a value dependent on α (Fig. 8a). For α values below 4°, the increase was nearly immediate. A further increase in α to 4° produced a substantial change in the p(t) relationship with a switch in porosity lasting for approximately 1.4 s. The porosity of the material flowing through the volume of the converging orifice was approximately 83% for α ≤ 4° and 53% for α ≥ 5°. The seemingly slight increase in α from 3° to 4° and subsequently to 5° produced substantial changes in the behavior of the material. The limiting value of the half cone angle was α = α crit. = 4°. The porosity inside the corresponding volume of the hopper of the half cone angle α = 4° during the discharge was 53%, i.e. it was the same as the values of a dense flow obtained for the converging orifice with α > α crit. . The same tendency for changes in the porosity was observed for the insert with h = 12 mm and α crit. = 19.7º (Fig. 8b). In this case, the relationships were not as clear as for h = 100 mm due to relatively big scatter of data resulting from discrete nature of the process averaged over eight times lover volume. The comparison of the profiles of velocity V z of particles during discharge for the flat orifice, converging orifice, and hopper with the same α and d 0 (Fig. 9) explains the cause of the increase in the mass discharge rate through the converging orifice to values obtained for the hopper. For the converging orifice, at the level of the bottom edge of the orifice the particle velocity was approximately twice higher than the velocity of particles leaving the orifice (Fig. 9a). Figure 8a shows that the porosity in the converging orifice was also approximately twice higher than in the hopper. Therefore, the mass discharge rate, the product of the particle velocity and the bulk density, was similar for the converging orifice and the hopper with the same half cone angle α.
Profiles of the particles velocity V z in the vertical direction have shown that the highest particles acceleration occurred in the converging orifice (Fig. 9b). Increase in the porosity during commencement of discharge through the converging orifice softened the structure of the bulk of particles, and, consequently, made accelerating particles easier due to gravity. Finally, it resulted in higher velocity at the level of the bottom edge of the orifice. Softening of the structure of the bulk of particles in the volume of the converging orifice with a thickness of a few particle diameters ensures the same mass discharge rate as the discharge of the dense structure of the bulk of particles through the hopper. This means that, by applying different geometries of the orifice, a similar mass discharge rate can be achieved by means of a stream of more densely packed particles with a lower particle velocity or a stream of more loosely packed particles with a higher particle velocity.

Discussion
The need of deeper understanding of the kinematic transition region near the outlet in the silo is important for a precisely controlled discharge rate 1,5 . Therefore, the orifice dimensions were selected as variables to study discharge through the converging orifice. The converging orifice can be considered as an extremely simplified curved hopper reduced into two segments: a flat floor and a short part of the hopper. Studies on the effect of the geometry of a conical converging orifice on the mass flow rate of granular material is scarce. Therefore, in this project, the results obtained for silos with conical hoppers were considered as a reference point. In the majority of cases, the flow rate through the converging orifice is higher than through the hopper with the same orifice diameter. Hence, the conical hopper may be replaced by flat bottom equipped with converging orifice with a smaller diameter to obtain the same discharge rate. The values of the MDR obtained for the converging orifice were located close to these provided by the hopper and considerably lower than the values provided by curved hopper, presented by Huang et al. 16,26 and Guo et al. 14 .
The main novelty of the study is the indication of the hyperbolic type relationship between the half cone angle α crit. and the thickness of the insert with converging orifice h separating the geometry of the converging orifice into two regions with respect of the dependence of the MDR on α for d 0 = const.: (1) the MDR increasing with α increase for α ≤ α crit. and (2) the MDR decreasing with α increase for α > α crit. . The results of this study corroborated the observation that the flow mode (bulk density of the stream and particle velocity) of granular material through a conical converging orifice depends on the half cone angle of the orifice. For α < α crit. , the discharge commencement produces a rapid increase in the porosity of the material in the volume of the orifice associated with the higher particle velocity. Attaining α = α crit. produced a substantial change. The increase in porosity with the discharge time was much slower and nearly linear. Slight surpassing α crit. (by one or two degrees) allowed a denser flow with a lower particle velocity.
At the flat floor of the bin, a dead zone is formed generating a natural hopper. In this area, the flow direction changes from vertical to converging, which is associated with softening of structure of the material. In a hopper, the change in the direction of particle movement is much smoother, which results in much lower dilation and acceleration along the straight line of particle movement. Despite such a big difference in the characteristics of particle movement between the converging orifice and the hopper, the mass discharge rate may be similar for the same half cone angle and appropriately adjusted height of the converging orifice. As concluded by Gella, Maza, & Zuriguel 8 , it is difficult to definitively state which specific property of particles is responsible for the macroscopic changes observed in the system. The relationship among all these magnitudes is not trivial, and further research is necessary to clarify these questions. Understanding how to manipulate and control the mass discharge rate may have a positive impact on the productivity and quality of industrial unit operations.

Conclusions
The following detailed conclusions were drawn: 1. Material discharges in the dense (α > α crit. , porosity ≈ 60%) or loose (α ≤ α crit. , porosity ≈ 80%) flow mode depending on the insert thickness h and the angle of inclination of the generatrix of the converging orifice α. The maximal normalized mass discharge rate MDR norm. decreased from 3.2 for h = 100 mm and α = 4º to www.nature.com/scientificreports/ 1.2 for h = 1.5 and α = 55º. In the majority of cases, the flow rate through the converging orifice is higher than through the hopper with the same orifice diameter. 2. For d 0 = const. the critical value of the half cone angle α crit. depended only on the insert thickness h. For α ≤ α crit. the mass discharge rate followed Beverloo's relationship obtained for the flat orifice. The hyperbolic type dependence of the critical value of the half cone angle α crit. on the insert thickness separated the geometry of the converging orifice (h,α) into two regions of opposite reaction of the mass discharge rate MDR to α increase: (1) increase of the MDR with α increase for α < α crit. and, (2) decrease of the MDR with α increase for α > α crit. . 3. The tendencies observed for the monodisperse assembly of spherical particles were preserved when beddings of wheat, lineseeds, and rapeseeds were tested. However, closer convergence of the results of the experiments and simulations would require fine tuning of the simulation parameters. The geometrical and mechanical parameters of real particles are far from those of a perfect sphere, which results in this discrepancy. 4. The results of the reported study show that the application of proper orifice geometry may allow precise control of the flow rate of granular material discharged from the silo. The fairly close compliance between the results of the experimental measurements and the simulations shows that DEM can be used to design equipment in systems involving granular flow.

Data availability
All data generated or analyzed during this study are included in this published article. Further detailed information on the datasets elaborated during the current study is available from the corresponding author and can be provided on reasonable request.